
linearly independent functions of the form \(\sum _{i=1}^k\alpha _it^{b_i}\)  where \(\alpha _1,\ldots , \alpha _k\in {\mathbb {Q}}\)  and \(b_1,\ldots , b_k \in (0,+\infty )\)  (this generalizes the main result of [1]}, which covers the case of integer polynomials), or, more generally,

functions of the form \(\sum _{i=1}^k\alpha _it^{b_i}\)  where \(\alpha _1,\ldots , \alpha _k\in {\mathbb {R}}\)  and \(b_1,\ldots , b_k\in (0,+\infty )\) , such that every linear combination of \(a_1,\ldots , a_\ell \) , with at least one coefficient irrational, is not a rational polynomial.Linear independence does not suffice in the case of real coefficients, to see this, take \(k=1\)  and \(a_1(t)=\alpha t\)  where \(\alpha \)  is irrational, then the sequence \([a_1(n)]\)  is not ergodic for the (totally ergodic) system \((m_ R)\) , where \(Rx=x+1/\alpha \) , \(x\in . \)

